Other ways to find line of "best" fit The most common methods I've seen to find a line of best fit are Least Squares regression and median-median. Are there other good ways? Is there a way to minimize the absolute value difference and find a line of best fit that way? Or to find the distance straight to a line instead of the vertical distance to the line? Thoughts?
 A: Minimizing the sum of absolute differences is quite common, as Nick Cox suggests, it's often called L1 regression or Least absolute deviations regression; it's also a specific case of quantile regression and many posts here relate to it. 
http://en.wikipedia.org/wiki/Least_absolute_deviations
http://en.wikipedia.org/wiki/Quantile_regression
The orthogonal distance (what I assume you mean by "straight-line distance") would correspond to a particular case of Deming regressing, itself a particular case of the total least squares line, called orthogonal regression, which will give the line of the first principal component. 
https://en.wikipedia.org/wiki/Principal_component_analysis
http://en.wikipedia.org/wiki/Deming_regression
http://en.wikipedia.org/wiki/Total_least_squares
There are many, many other lines that might be fitted; a couple of examples include  Theil-Sen regression or more generally, robust regression, which includes many different techniques.
Some discussion of robust regression (including some comparison of Theil-Sen and L1 regression) is here.
There's some interesting discussion relating correlation measures to straight-line fits here
